On the multifractal spectrum of weighted Birkhoff averages

<p style='text-indent:20px;'>In this paper, we study the topological spectrum of weighted Birk–hoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave over its domain. In case of typical weights with respect to some ergodic quasi-Bernoulli measure, we determine the spectrum. Moreover, in case of full shift and under the assumption that the potentials depend only on the first coordinate, we show that our result is applicable for regular weights, like Möbius sequence.</p>

Strongly aperiodic subshifts of finite type on hyperbolic groups

We prove that a hyperbolic group admits a strongly aperiodic subshift of finite type if and only if it has at most one end.

Asymptotic Counting in Conformal Dynamical Systems

In this monograph we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the former. We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic orbits ordered by a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features of the distribution of their weights. These results have direct applications to a wide variety of examples, including the case of Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions, Mannenville-Pomeau maps, Schottky groups, Fuchsian groups, and many more. This gives a unified approach which both recovers known results and proves new results. Our new approach is founded on spectral properties of complexified Ruelle–Perron–Frobenius operators and Tauberian theorems as used in classical problems of prime number theory.

Nilpotent endomorphisms of expansive group actions

We consider expansive group actions on a compact metric space containing a special fixed point denoted by [Formula: see text], and endomorphisms of such systems whose forward trajectories are attracted toward [Formula: see text]. Such endomorphisms are called asymptotically nilpotent, and we study the conditions in which they are nilpotent, that is, map the entire space to [Formula: see text] in a finite number of iterations. We show that for a large class of discrete groups, this property of nil-rigidity holds for all expansive actions that satisfy a natural specification-like property and have dense homoclinic points. Our main result in particular shows that the class includes all residually finite solvable groups and all groups of polynomial growth. For expansive actions of the group [Formula: see text], we show that a very weak gluing property suffices for nil-rigidity. For [Formula: see text]-subshifts of finite type, we show that the block-gluing property suffices. The study of nil-rigidity is motivated by two aspects of the theory of cellular automata and symbolic dynamics: It can be seen as a finiteness property for groups, which is representative of the theory of cellular automata on groups. Nilpotency also plays a prominent role in the theory of cellular automata as dynamical systems. As a technical tool of possible independent interest, the proof involves the construction of tiered dynamical systems where several groups act on nested subsets of the original space.

Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type

<p style='text-indent:20px;'>In this text I study the asymptotics of the complexity function of <i>minimal</i> multidimensional subshifts of finite type through their entropy dimension, a topological invariant that has been introduced in order to study zero entropy dynamical systems. Following a recent trend in symbolic dynamics I approach this using concepts from computability theory. In particular it is known [<xref ref-type="bibr" rid="b12">12</xref>] that the possible values of entropy dimension for d-dimensional subshifts of finite type are the <inline-formula><tex-math id="M1">\begin{document}$ \Delta_2 $\end{document}</tex-math></inline-formula>-computable numbers in <inline-formula><tex-math id="M2">\begin{document}$ [0, d] $\end{document}</tex-math></inline-formula>. The kind of constructions that underlies this result is however quite complex and minimality has been considered thus far as hard to achieve with it. In this text I prove that this is possible and use the construction principles that I developped in order to prove (in principle) that for all <inline-formula><tex-math id="M3">\begin{document}$ d \ge 2 $\end{document}</tex-math></inline-formula> the possible values for entropy dimensions of <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional SFT are the <inline-formula><tex-math id="M5">\begin{document}$ \Delta_2 $\end{document}</tex-math></inline-formula>-computable numbers in <inline-formula><tex-math id="M6">\begin{document}$ [0, d-1] $\end{document}</tex-math></inline-formula>. In the present text I prove formally this result for <inline-formula><tex-math id="M7">\begin{document}$ d = 3 $\end{document}</tex-math></inline-formula>. Although the result for other dimensions does not follow directly, it is enough to understand this construction to see that it is possible to reproduce it in higher dimensions (I chose dimension three for optimality in terms of exposition). The case <inline-formula><tex-math id="M8">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula> requires some substantial changes to be made in order to adapt the construction that are not discussed here.</p>